Singular extremals in multi-input time-optimal problems: a sufficient condition

In this paper we study second order sufficient conditions for the strong-local optimality of singular Pontryagin extremals. In particular, we focus on the minimum-time problem for a control-affine system with vector inputs. We use Hamiltonian methods to prove that the coercivity of a suitably-defined second variation - plus an involutivity assumption on the distribution of the controlled fields - is a sufficient condition for the strong optimality of a candidate extremal.